(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will think of as a category and also as a directed graph (objects = vertices, morphisms = edges). Assume we have a functor $F$ from the graph into (say) chain complexes. We will construct a big chain complex (the homotopy colimit) in stages.

Stage 0: direct sum over all vertices $v$ of $F(v)$

Stage 1: direct sum over all edges $e$ of the mapping cylinder of $F(e)$, with the ends of the mapping cylinder identified with the appropriate parts of stage 0.

Stage 2: direct sum over all pairs of composable edges $(e_1, e_2)$ of a higher order mapping cylinder, with appropriate identifications to parts of stage 1. This implements a relation between the three stage 1 mapping cylinders corresponding to $e_1, e_2$ and $e_1*e_2$.

(1) I happen to like this paper, but of course I'm biased. (I hope self-citation isn't forbidden here...) However, I started writing that paper mainly because I couldn't find an existing reference/introduction that I liked. So if someone has another reference to suggest I would love to hear about it.

(2) This sounds like the simplicial bar construction, which is the one I used in my paper above. I think I included some other references in the bibliography.